Sobolev spaces. Elliptic equations

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1 Sobolev spaces. Elliptic equations Petru Mironescu December Introduction The purpose of these notes is to introduce some basic functional and harmonic analysis tools (Sobolev spaces, singular integrals) and to explain how these tools are used in the study of elliptic partial differential equations. In a last part, we will introduce some basic variational methods, applied to existence of solutions of semi linear elliptic problems. Many books and papers are at the origin of these notes: Sobolev spaces Robert A. Adams, John J.F. Fournier: Sobolev Spaces. 2nd ed, Elsevier 2003 Haïm Brezis, Analyse fonctionnelle. Théorie et applications, Masson 1983 Michel Willem, Analyse fonctionnelle élémentaire, Cassini 2003 Louis Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa 13 (1959), p Haïm Brezis, Augusto C. Ponce, Kato s inequality up to the boundary, Comm. Contemp. Math. 10 (2008), p Lars Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, 1990 Richard Courant, David Hilbert, Methods of Modern Mathematical Physics, II, Interscience, 1962 Elliott H. Lieb, Michael Loss, Analysis, 2nd edition, American Mathematical Society, 2001 Moshe Marcus, Victor Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979), no. 2, Xavier Lamy showed me a proof, much simpler than the one I initially found, of Lemma I included his proof in the text 1

2 Singular integrals Elias Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993 Elias Stein, Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, 1971 Lawrence Evans, Ronald Gariepy, Measure theory and fine properties of functions, CRC Press, 1992 Linear elliptic equations David Gilbarg, Neil S. Trudinger, Elliptic partial differential equations of second order, 4th ed., Springer 2001 Quin Han, Fanghua Lin, Elliptic Partial Differential Equations, American Mathematical Society Notations a) If 1 p, p stands for the conjugate of p (thus 1 p + 1 p = 1) b) N is the space dimension c) R N + = {x R N ; x N > 0}, R N = {x R N ; x N < 0} d) Ω is an open set in R N. Unless stated otherwise, Ω is supposed connected (i. e., Ω is a domain) e) α, β stand for multi-indices in N N. We let α = N α j j=1 f) A is the (usually Lebesgue, sometimes Hausdorff) measure of A g) denotes the average integral: f = 1 f A A A ( u h) stands for the standard Euclidean norm. E. g., u = x j i) ρ is a standard mollifier, i. e. a map s. t. ρ Cc (R N ; R), ρ 0, ρ = 1, supp ρ B(0, 1) j) ν stands for the unit outward normal at the boundary of a smooth domain Ω 2 ) 2

3 k) A Ω means that A is a compact subset of the open set Ω l) In principle, K is always a compact set, but I may forget this here and there and let K also denote a constant. Also in principle, ω is an open subset of Ω m) The subscript loc stands for the local version of the spaces we consider. This will not be defined each time, so that we content ourselves to give, once for all, an example: W 1,1 loc (Ω) = {u L1 (Ω); u K L 1, u K L 1, K Ω} n) We let ω N denote the volume of the (Euclidean) unit ball in R N, and σ N denote the (H N 1 dimensional) measure of the (Euclidean) unit sphere. Recall that these two quantities are related by σ N = Nω N o) M is the set of measurable functions p) If C is a ball in R N (with respect to some norm), then we let C denote the cube having the same center as C and twice the same radius. Similarly, C has four times the radius of C 1 Sobolev spaces 1.1 Motivation Let u solve the problem { u = f in Ω u = 0 on Ω, (1.1) aka as the Dirichlet problem for the Poisson equation. Assume everything smooth. Multiply the first equation by v C 2 (Ω) s. t. v = 0 on Ω and integrate once by parts (i. e., use the first Green formula). We find that Ω u v i. e., u v Ω Ω Ω u ν v = Ω fv, fv = 0, v X, 3

4 where X := {v C 2 (Ω); v = 0 on Ω}. Formally, this is the same as DJ(u) = 0, where J : X R is given by J(u) := 1 u 2 fu. 2 Ω This suggests the following strategy intended to solve (1.1): minimize J in X. Then the minimizer should solve (1.1). Good news: if u is a minimizer, then it is indeed the solution of (1.1). Problem with that: the minimum need not exist. A first reason is that if we take a minimizing sequence (i. e., a sequence such that J(u n ) min), then there is no reason to obtain some u X s. t. u n u (whatever the sense we give to this convergence). Actually, Weierstrass showed that, if f is continuous, then there may not be a minimum point of J in X. A way to overcome this difficulty is to replace X by a larger space, where the minimum is attained (but not necessarily by some u X). It turns out that the good candidate is the closure Y of X for ( ) 1/2 the norm u u 2. As we will see later, this is a Sobolev space. Ω Sobolev spaces are useful even when (1.1) does have a solution in X. Indeed, existence can be proved as follows: start by minimizing J in Y (this is always possible), then prove that u X (this requires extra assumptions on f, and the proof of such type of results is the purpose of regularity theory). This is Hilbert s strategy. 1.2 Distributions Distributions were formalized by Schwartz, but predecessors of this theory appear already in the works of Leray and Sobolev. 1.1 Definition. Let Ω R N be an open set. A distribution on Ω is a linear functional u on Cc (Ω) which is continuous in the following sense: for each compact K Ω, there is a constant C and an integer k s. t. u(ϕ) C sup α ϕ, ϕ Cc (Ω) s. t. supp ϕ K. α k The vector space of the distributions is denoted D (Ω). 1.2 Example. A continuous function u defines a distribution, still denoted u, through the formula u(ϕ) = ϕ. More generally, one can replace continuous Ω by measurable and integrable on compacts. In this case, we may take k = 0 and C = u. K 4 Ω

5 There is a good reason to keep the notation u for the above distribution 1.3 Proposition (Localisation principle). Let u, v L 1 loc (Ω). Then u = v a. e. iff the distributions defined by u and v are equal. In other words, one can identify u with the associated distribution. Proof. The only if part is clear. For the if part, we rely on the following fact that we will prove later 1.4 Proposition. Let ρ be a standard mollifier. Let f L 1 loc (Ω). Then f ρ ε f a. e. as ε Remark. Note that f ρ ε is defined in Ω ε = {x Ω; dist (x, Ω) > ε}. Thus, for each x Ω, f ρ ε (x) is well defined for small ε (smallness depending on ε). Back to the proof of the localisation principle. With f = u v, we have fϕ = 0 for each ϕ C c conclude by letting ε 0. (Ω). In particular, f ρ ε = 0 for each ε. We 1.6 Example. If a Ω, then δ a (ϕ) = ϕ(a) is a distribution (the Dirac mass at a). Indeed, we may take k = 0 and C = 1. When a = 0, we write δ rather than δ Example. If Σ is a k-dimensional submanifold of Ω, then δ Σ (ϕ) = ϕ dh k is a distribution (the Dirac mass on Σ). Indeed, we may take Σ k = 0 and C the Hausdorff measure of Σ K. More generally, we may consider the distribution fh k Σ, where f is locally integrable (with respect to H k ) on Σ. This distribution acts through the formula ϕ fϕ dh k. Σ 1.8 Example. All the above examples are special cases of measures: if µ is a locally finite Borel measure, then µ defines a distribution through the formula µ(ϕ) = ϕ. (Take k = 0 and C = µ (K).) Ω 1.9 Exercise. It is not always possible to take k = 0. For example, let u(ϕ) = ϕ (0) (assuming that 0 Ω and N = 1). Then u D (Ω), but it is not possible to take k = 0 when, say, K is a closed interval centered at the origin. Hint: consider ϕ(nx) for a fixed ϕ. The next result identifies (positive) measures. 5

6 1.10 Proposition. Let u D (Ω). Then the following are equivalent: (i) u is positive, in the sense that u(ϕ) 0 if ϕ Cc (Ω) and ϕ 0 (ii) There is some positive Radon measure µ s. t. u(ϕ) = ϕ dµ, ϕ C c (Ω). If these conditions are satisfied, we write u 0. Proof. The implication (ii)= (i) is clear. Conversely, let K Ω and let ψ C c (Ω) be s. t. 0 ψ 1 and ψ 1 in K. If ϕ C c (K), then ϕ L u(ψ) u(ϕ) ϕ L u(ψ), so that u(ϕ) C(K) ϕ L, ϕ Cc (K). We find that u has a linear positive continuous extension to C c (Ω). We conclude via the Riesz representation theorem. When u C 1 (Ω), we have j u(ϕ) = u( j ϕ) (this is obtained by integration by parts). This suggests the following 1.11 Definition. If u D (Ω), we define j u through the formula j u(ϕ) = u( j ϕ). More generally, α u(ϕ) = ( 1) α u( α ϕ). This is still a distribution. If P ( ) = a α α, then we define (P ( )u)(ϕ) = ( 1) α a α u( α ϕ). In the same vein, we define, for u D (Ω) and a C (Ω), the distribution au by (au)(ϕ) = u(aϕ). Two points are of interest: the result is still a distribution, and these definitions coincide with the usual ones when u is smooth enough. For example, when u C 1, we have old j u=new j u. When we want to emphasize such an equality, we write j,p u = j,d u (p stands for point derivative, d for distributional derivative) Remark. When there is a possible doubt, we let the subscript p stand for point quantities, and d for distributional ones Exercise. If a C (Ω) and u D (Ω), we have the following Leibniz rule α (au) = α! β!(α β)! β a α β u. β α 1.14 Exercise. If f(x) = x (in R), then f = sgn. 6

7 1.15 Exercise. If f(x) = { 1, if x > 0 0, if x 0 (in R), then f = δ Exercise. Let Ω 1, Ω 2 be smooth open sets s. t. Ω 1 Ω 2 = and Ω 1 Ω 2 = { Σ, with Σ smooth hypersurface. Let u i C 1 (Ω i Σ), i = 1, 2, and u 1, in Ω 1 set u =. Then j u = j u 1 χ Ω1 + j u 2 χ Ω2 + (u 2 u 1 )ν j H k Σ. u 2, in Ω 2 Here, ν is the normal to Σ directed from Ω 1 to Ω 2. A slightly more involved result is the following 1.17 Proposition. Assume that N 2, 0 Ω. Let f C 1 (Ω \ {0}) be such that p u L 1 loc (Ω). Then j,pu = j,d u. Proof. We have to prove that the subscript p. The key fact is that Ω u j ϕ = S(0,ε) Ω j,p uϕ, ϕ C c (Ω). We drop u dh N 1 = o(1) as ε 0. (This will imply u L 1 loc (Ω).) Assuming this for the moment, we argue as follows Ω ( u j uϕ = lim j uϕ = lim ε 0 Ω\B(0,ε) ε 0 = lim u j ϕ = ε 0 Ω\B(0,ε) B(0,1)\B(0,ε) Ω ν j uϕ S(0,ε) u j ϕ. Ω\B(0,ε) ) u j ϕ We now prove the key fact (and its consequence). Assume e. g., in what follows, that B(0, 1) Ω. We start by noting that (by dominated convergence) ε N 1 lim u = 0. ε 0 x N 1 We claim that u Cε N 1 + S(0,ε) This follows from u = ε N 1 S(0,ε) B(0,1)\B(0,ε) ε N 1 u. x N 1 u(εω) dh N 1 S N 1 1 = ε N 1 u(ω) d S N 1 ε dr [u(rω)] N 1 dh ε S N 1 u dh N 1 ε N 1 + u. x N 1 N 1 7 Ω\B(0,ε)

8 Consequence: if K Ω, then u u + sup u sup K\B(0,1) 0 ε 1 K B(0,1) i. e., u L 1 loc (Ω). S(0,ε) u + sup u <, K\B(0,1) In order to establish our next example, we need the following simple fact 1.18 Proposition (Delocalisation principle). If u = v on Ω i, i I, then u = v on Ω i. Proof. Let Ω = Ω i, ϕ Cc (Ω) and ϕ i be a finite partition of unity on supp ϕ subordinated to the covering (Ω i ). Then u(ϕ) = u(ϕ i ϕ) = v(ϕ i ϕ) = v(ϕ) Proposition. Let Σ be a k-dimensional submanifold of Ω (here, k N 2). Let u C 1 (Ω \ Σ) be such that p u L 1 loc (Ω). Then j,pu = j,d u. Conversely, if j,d u L 1 loc (Ω), then j,pu L 1 loc (Ω). Note that Proposition 1.17 is a special case (when k = 0) of the above result. Proof. We start with a special case, later referred as the standard case. We let Ω = B R N k(0, 1) ( 1, 1) k, Σ = {0} ( 1, 1) k and assume that u C 1 (Ω\Σ). Write a point in R N as x = (x, x ), with x R N k, x R k. As in the proof of Proposition 1.17, we have u Cε N k 1 ε N k 1 + u 0 as ε 0. x N k 1 {x Ω; x =ε} Consequently, we have u L 1 (Ω) and u = o(ε) as ε 0. {x Ω; x ε} Let now ψ C (R; R) be s. t. ψ(t) = have Ω u j ϕ = lim = = ε 0 Ω Ω Ω Ω { 1, if t 1 0, if t 1/2. For ϕ C c (Ω) we uψ( x /ε) j ϕ = lim j (uψ( x ε 0 Ω /ε))ϕ j uϕ lim ε 0 Ω 1 j uϕ lim ε 0 ε O u j [ψ( x /ε))]ϕ ( ) 8 u {x Ω; x ε} = j uϕ. Ω

9 We now turn to the general case. We cover Ω with a family (Ω i ) of open sets s. t., for each i: either Ω i Σ =, or there is a C 1 diffeomorphism Φ i : Ω i Ω 0 (here, Ω 0 stands for the open set from the standard example) s. t. Φ(Σ Ω i ) = {0} R k. In view of the delocalisation principle, it suffices to prove that j,p u = j,d u in each Ω i. This equality is clear if Σ does not meet Ω i. Otherwise, let ϕ Cc (Ω) be s. t. supp ϕ Ω i. Then u j ϕ = lim uψ( (Φ i (x)) /ε) j ϕ = lim j (uψ( (Φ i (x)) /ε))ϕ Ω ε 0 ε 0 i Ω i = uϕ lim u j (ψ( (Φ i (x)) /ε))ϕ = uϕ. Ω ε 0 i Ω i Ω i Here, we use the fact that j (ψ( (Φ i (x)) /ε)) C/ε combined with u C u Φ 1 i = o(ε) as ε 0. {x Ω i ; (Φ i (x)) ε} {x Ω 0 ; x ε} The last property is a consequence of the fact that u Φ 1 i falls into the standard case. We end with the converse: since u C 1 (Ω \ Σ), we have j,p u = j,d u a. e. in Ω \ Σ. Since Σ is a null set, we find that j,p u = j,d u a. e. in Ω, so that j,p u L 1 loc. Another useful operation (in addition to differentiation and multiplication be smooth functions) is convolution Definition. If u D (R N ) and ϕ C c (R N ), we set u ϕ(x) = u(ϕ(x )). One may prove the following 1.21 Proposition. We have u ϕ C and β+γ (u ϕ) = ( β u) ( γ ϕ). Proof. The second property is trivial. We sketch the argument leading to the first one. The key fact is continuity, which is obtained as follows: if x n x, then there is a fixed compact K s. t. supp ϕ(x n ) K for each n. Since α (ϕ(x n ) ϕ(x )) 0 uniformly in K, we find (using the definition of a distribution) that u ϕ(x n ) u ϕ(x). Similarly, we have u ϕ(x + te j ) u ϕ(x) lim t 0 t = u j ϕ(x). By the key fact, the latter quantity is continuous in x. Thus u ϕ C 1 and j u ϕ = u j ϕ. We continue by induction on the number of derivatives. 9

10 1.22 Example. δ ϕ = δ. C c Convolution can be considered in other settings (not only D (R N ) vs (R N )), e. g., L 1 vs L p. We will come back to this later Exercise. A final point we discuss here is extension. If u D (Ω) and ω is an open subset of Ω, then we may restrict u to ω in the obvious way: we let u act on Cc (ω). The converse is not always possible: given a distribution u on ω, it may not be the restriction of a distribution on Ω. Check the following: if ω = (0, 2) and Ω = R, there is no distribution on Ω whose restriction to ω is u = δ 1/n. 1.3 First properties of Sobolev spaces Definition 1.24 Definition. Let 1 p. Then W 1,p = W 1,p (Ω) = {u L p (Ω); j u L p (Ω)}. We endow this vector space with the norm u W 1,p When p <, another possible (equivalent) norm is When p = 2, W k,2 is aka H 1. ( = u L p + j u p L p + j j u L p. j u p L p ) 1/p. Note that L p L 1 loc, so that Lp functions are in D (Ω). Thus j u makes sense (as a distribution) Example. Assume that Ω = B R N (0, 1). Let u(x) = x α, where α R. We claim that u W 1,p iff p(α + 1) < N. Indeed, to start with, we have u L p iff pα < N. Next, we have p u(x) x α 1, so that p u L 1 loc if and only if α < N 1. If α N 1, then we cannot have d u L 1 loc (in view of Proposition 1.19). Thus we cannot have u W 1,p. If α < N 1, then p u = d u, so that u L p iff p(α + 1) < N. We find that u W 1,p iff p(α + 1) < N W 1, and Lip 1.26 Theorem. We have u W 1, iff (possibly after redefining u on a null set) u is bounded and there is some C > 0 s. t. (P) u(x) u(y) C x y 10

11 whenever [x, y] Ω. In the special case where Ω is convex, this is the same as u bounded and Lipschitz. In this case, u W 1, u L + u Lip Remark. In particular, the theorem asserts that u equals a. e. a continuous function. We will write, here and there, u is continuous as a shorthand for u is equal a. e. to a continuous function. Proof. Assume first that u W 1,. Then (u ρ ε ) = ( u) ρ ε ) u L ρ ε L 1 = u L. We find that the family (u ρ ε ) satisfies u ρ ε u L and (by the mean value theorem) condition (P ). By Arzelà-Ascoli, we have (possibly after extraction) u ρ ε v uniformly on compacts. Clearly, the limit is continuous, bounded (since u is), and satisfies (P ). Since, on the other hand, we have u ρ ε u a. e., we proved the only if part. Conversely, by delocalisation we may assume Ω bounded and convex. We let as an exercise the fact that if (P ) holds, then u ρ ε is C-Lipschitz. Thus (u ρ ε ) C. Thus the family ( (u ρ ε )) is bounded in L (and thus in L 2 ). It is a standard fact in functional analysis that, under such assumptions, possibly after passing to a subsequence, we have (u ρ ε ) f in L 2 for some f s. t. f C. Using the definition of weak convergence, we find that d u = f. On the way, we also proved norm equivalence when Ω is convex Exercise. Prove the standard fact mentioned above, which amounts to: if f n, f L 2 (Ω) are s. t. f n C and f n f in L 2, then f C a. e. (here, f n, f may be vector-valued). f Hint: compute f f. f >C 1.29 Exercise. Let Ω = {x R 2 \ R ; 1 < x < 2} and u(re ıθ ) = θ, 1 < r < 2, θ ( π, π). Prove that u W 1,, but u Lip. We will now see for the first time the role of the regularity of Ω. In view of the above example, in general we do not have W 1, = Lip. However, this holds under additional assumptions on Ω. A deep question that will be systematically overlooked in what follows is the one of minimal assumptions that make useful theorems (embeddings, compactness, etc.) work. Most of time, we will consider lazy assumptions that make some rather simple proofs work. For sharper results in this direction, the books of Adams and Fournier, respectively Maz ja, on Sobolev spaces, are good references. 11

12 1.30 Theorem. Assume that Ω is a bounded Lipschitz domain. Then W 1, = Lip, and u W 1, u L + u Lip. Proof. When Ω is Lipschitz, the geodesic distance d Ω in Ω is equivalent to the Euclidean one. (Recall that d Ω (x, y) is the infimum of the length of all polygonal lines connecting x to y. Such lines do exist, since Ω is a domain.) Clearly, the proof of Theorem 1.26 implies that u W 1, u L + N u Lip. Conversely, by the same theorem we have u(x) u(y) u L d Ω (x, y). We conclude using the fact that the geodesic distance is of the same order as the Euclidean one Exercise. Prove the equivalence between geodesic and Euclidean distance in Lipschitz bounded domains. Hint: cover Ω with balls which are: either contained in Ω, or chart domains on which Ω can be straightened. Extract a finite covering, then consider the Lebesgue number r associated to this covering. Estimate d Ω (x, y) by considering the cases x y r and x y > r D We assume, e. g., that 0 Ω and that Ω is an interval. The description of Sobolev spaces is basically a consequence of the following 1.32 Theorem. Let u, v L 1 loc ( 1, 1). Then u = v iff there is some C s. t. (possibly after redefining u on a null set) u(x) = C + x 0 v(t) dt. Proof. It is straightforward (by definition+fubini) that u 0 (x) = satisfies u 0 = v. The conclusion follows from the next lemma. x 0 v(t) dt 1.33 Lemma. Let u D (Ω) (with Ω R an interval). Then u = 0 iff u L 1 loc and u = C a. e. Proof. The if part is clear. Conversely, fix ϕ 0 Cc (Ω) s. t. ϕ 0 = 1. Let ( ) ϕ Cc (Ω) ans set ψ = ϕ ϕ ϕ 0. Then ψ has zero average, which implies that ψ = ζ for some ζ Cc (Ω). We find that, with C = u(ϕ 0 ), we have u(ϕ) = Cϕ + u(ψ) = Cϕ u (ζ) = Cϕ, i. e., u = C in D (Ω). We conclude via the localisation principle. 12

13 1.34 Remark. From now on we identify maps with derivatives in L 1 loc with their (existing and unique) continuous representative Corollary. If Ω is bounded, then W 1,p C(Ω), with continuous embedding. Proof. Inclusion is clear. Continuity follows from the closed graph theorem (since, as we will se later, W 1,p is a Banach space) Corollary. Assume that u L 1 loc. Then u has a (usual) derivative u p a. e., and u p = u a. e. Proof. Let u = v. Then, for a. e. x Ω, we have (by the Lebesgue differentiation theorem) u p(x) = lim h 0 x+h x v(t) dt h = v(x) Corollary. Lipschitz functions of one variable are differentiable a. e. The next result is due to Lebesgue Theorem. Assume that u is bounded. Then u W 1,1 iff u is absolutely continuous. Proof. Recall that absolutely continuity means: for each ε > 0, there is some > 0 s. t. if (a i, b i ) are disjoint intervals in Ω s. t. (bi a i ) <, then u(b i ) u(a i ) < ε. The only if condition is a special case of Lebesgue s lemma applied to v = u. This lemma asserts that if A <, then v < ε (provided is sufficiently small). If we take A = (a i, b i ), then we recover the absolute continuity condition. Conversely, if u is absolutely continuous (AC), then the following facts are easy to check and left as an exercise: a) If u is AC, then u is continuous and has bounded variation b) If we let u = u 1 u 2 be the Jordan decomposition of u (i. e., u 1 (x) = x 0 u, u 2 = u 1 u; these functions are non decreasing), then u 1, u 2 are AC c) Write each u i, i = 1, 2, as u i (x) = C i + µ i ((, x)) for appropriate measures µ i. (This is possible since each u i is continuous and non decreasing.) Then µ i has the property that if ω is an open set and ω <, then µ i (ω) < ε 13 A

14 d) Consequently, if ω is a Borel null set, then µ i (ω) = 0 e) By the Radon-Nikodym theorem, there is some v i L 1 s. t. u i (x) = D i + x 0 v i (t) dt f) This implies at once that u = v 1 v 2 L Theorem. The following are equivalent, when 1 p < : a) u = 0 on Ω b) u belongs to the closure, in W 1,p, of C c (Ω) c) the extension of u with the value 0 outside Ω is in W 1,p (R). Proof. We assume that Ω is bounded, e. g., Ω = ( 1, 1). The proofs have to be adapted to the other cases (Ω is a half line or R). (a) implies (c). Let denote the extension with the value zero outside Ω. Let v = u. Then 1 1 v = 0. We find that ũ(x) = x 1 ṽ(t) dt. This implies (c). (c) implies (b). Let w = ũ. We clearly have w = 0 outside Ω and w = 0. Let (w n ) Cc (Ω) be s. t. w n w in L p and w n = 0. Set u n (x) = x 1 w n (t) dt. Then u n C c (Ω). We leave as an exercise that u n u in W 1,p. (b) implies (a). Note that, if u 1, u 2 W 1,p (Ω) have derivatives v 1, v 2 and if u i ( 1) = 0, i = 1, 2, then u 1 (x) u 2 (x) v 1 v 2 L 1 C v 1 v 2 L p. With this in mind, a Cauchy sequence (in W 1,p ) (u n ) Cc (Ω) converges uniformly. In particular, we must have u( 1) = 0 (and, similarly, u(1) = 0.) n dimensional case: basic properties 1.40 Proposition. W 1,p is a Banach space. Proof. Clearly, if (u n ) is a Cauchy sequence, then u n u in L p and j u n v j in L p for appropriate u and v j. It is obvious that v j = j u. Finally, we clearly have u n u in W 1,p. 14

15 In the same vein 1.41 Proposition. W 1,2 is a Hilbert space (with the second norm) Proposition (Approximation by regularization). Assume that either Ω = R N or u vanishes outside some compact subset of Ω. Let 1 p < and u W 1,p. Then u ρ ε u in W 1,p. Proof. This is straightforward using the fact that j (u ρ ε ) = ( j u) ρ ε Exercise. Let u 0 (x) = x, x ( 1, 1). Let ψ C c ( 1, 1) be s. t. ψ = 1 near the origin. Let u = u 0 ψ. Then u W 1,, u is compactly supported, but there is no sequence of smooth maps converging to u in W 1,. In fact, prove that the closure of smooth maps consists precisely in C 1 maps Proposition (Approximation by cutoff and regularization). Let 1 p <. Then C c (R N ) is dense in W 1,p (R N ). Proof. It suffices to prove that Cc (R N ) is dense in C W 1,p (R N ) (next apply the previous proposition). Let u C W 1,p. Let ϕ Cc (R N ) be s. t. 0 ϕ 1, ϕ = 1 in B(0, 1), ϕ = 0 outside B(0, 2). If we set u ε = uϕ(ε ) Cc (R N ), then u u ε W 1,p u L p (R N \B(0,1/ε))+Cε u L p+ u L p (R N \B(0,1/ε)) 0 as ε 0. The next result is due to Meyers and Serrin Theorem. Assume that 1 p <. Then C (Ω) W 1,p is dense in W 1,p. Proof. Consider a sequence (Ω i ) of open sets s. t. Ω 1 =, Ω i Ω i+1 and Ω i = Ω. Let (ζ i ) be a partition of unity subordinated to the covering (Ω i+1 \ Ω i 1 ). Let u W 1,p. Using approximation by regularization, for each i there is some w i C s. t. supp w i Ω i+1 \ Ω i 1 and w i ζ i u W 1,p < 2 i 1 ε. We now let w = w i. Then w C, since in the neighbourhood of a point at most four w i s do not vanish. If ω Ω, we find that w u W 1,p (ω) < ε. We conclude by letting ω Ω. 15

16 1.3.5 Higher order spaces 1.46 Definition. Let 1 p and k = 1, 2,... Then W k,p = W k,p (Ω) = {u L p (Ω); α u L p (Ω), α k}. We endow this vector space with the norm u W k,p = α u L p. When α k p <, another possible (equivalent) norm is α u p L p When p = 2, W k,2 is aka H k. α k Straightforward generalizations of the previous results include 1.47 Theorem. Assume that Ω is a Lipschitz bounded domain. Then W k, consists precisely of C k 1 maps s. t. D k 1 is Lipschitz Theorem. If 1 p <, then C (Ω) W k,p is dense in W k,p Proposition (Approximation by regularization). Assume that either Ω = R N or u vanishes outside some compact subset of Ω. Let 1 p < and u W k,p. Then u ρ ε u in W k,p Proposition (Approximation by cutoff and regularization). Let 1 p <. Then C c (R N ) is dense in W k,p (R N ). Another obvious result that holds for each k is 1.51 Proposition. Assume that a C k (Ω) has bounded derivatives up to the order k. If u W k,p (Ω), then au W k,p (Ω) and the usual Leibniz rule applies to the derivatives of au up to the order k. Proof. This is clear when u C W k,p. approximation. 1/p The general case follows by 1.52 Exercise. Let Ω 1, Ω 2 be smooth open bounded sets s. t. Ω 1 Ω 2 = and Ω 1 Ω 2 = Σ, with Σ smooth { hypersurface. Let u i C k (Ω i ), i = 1, 2, be u 1, in Ω 1 Σ s. t. u 1 = u 2 on Σ. Set u =. Assume that u C k 1. Then u 2, in Ω 2 { u W k,p, 1 p, and, for α k, we have α α u 1, in Ω 1 Σ u =. α u 2, in Ω 2 More generally, one may replace the condition u i C k (Ω i ) by u i C k (Ω i Σ) and α u i L p (Ω i ), α k. Hint: consider first the case k = 1, then reduce the general case to this one. 16.

17 1.53 Exercise. Let Ω, U be smooth bounded domains. Assume that Φ : Ω U is a C k diffeomorphism. Then u W k,p (Ω) iff u Φ 1 W k,p (U). Hint: prove that the chain rule holds for α (u Φ 1 ), α k. (Second hint: start with a smooth u.) Extensions, straightening It happens to W 1,p maps what happens to distributions: in general, it is impossible to extend a W 1,p { maps in Ω to a W 1,p map in R N. Indeed, let 1, if x > 0 Ω = ( 1, 1) \ {0}, u(x) = 0, if x < 0. Then u = 0, so that u W 1,1. However, u does not have a W 1,1 extension to R, and not even to ( 1, 1). Indeed, otherwise u would equal, a. e. in ( 1, 1), a continuous function, which is impossible. As in the case where we discussed equality W 1, = Lip, extension property holds if we impose some regularity on Ω. Before examining that point, let us examine a simple (but not optimal) procedure that reduces the case of an arbitrary domain to the standard case where Ω is a half space, Ω = R N +. This procedure is the following: in order to establish a property, say (P ), for Ω, do the following: a) establish (P ) for Ω = R N + b) establish, via diffeomorphism (=straightening of the boundary), (P ) in a neighbourhood of a given point of Ω c) conclude via a partition of unity. We will explain this in detail when (P ) is the extension problem. In other situations, we will concentrate ourselves on the heart of the matter, which concerns the standard case. The other steps are rather straightforward and will be left to the reader Theorem. Assume that 1 p <. Let Ω be a bounded C k domain. Then there is a linear continuous extension operator P : W k,p (Ω) W k,p (R N ). Here, extension operator means that (P u) Ω = u Remark. Assumption on Ω is not optimal; Lipschitz, and even less, would be sufficient. We send to Adams and Fournier for sharper statements Remark. The theorem is also true when p = (Whitney s extension theorem), but this requires a separate proof and will be omitted here. 17

18 Proof. Step 1. The standard case Ω = R N + Write a point x R N as x = (x, x N ); write also α N N as α = (α, α N ). u(x), if x N > 0 Let P u(x) = k a j u(x, jx N ), if x N < 0. Here, the real numbers a j will j=1 be fixed later. We claim that, for appropriate a js and α s. t. α k, we have α u(x), if x N 0 α P u(x) = k a j ( j) α N α u(x, jx N ), if x N < 0. (1.2) j=1 Assuming this, we leave to the reader the fact that P has all the required properties. We start with the special case where, in addition to being in W k,p, we assume that u C (R N +). In this case, we have P u C k 1 for appropriate a j. Indeed, the derivatives up to the order k 1 from above and k below R N 1 {0} will coincide if the a j s satisfy the system a j ( j) l = 1, l = 0,..., k 1. This (Vandermonde) system has a unique solution. E. g., when k = 1, a 1 = 1, and P is the extension by reflection across R N 1 {0}. By Exercise 1.52, (1.2) holds in this case. We now turn to the case of a general u. The conclusion is a straightforward consequence of the following 1.57 Proposition. Let 1 p <. Then C (R N +) W k,p (R N +) is dense in W k,p (R N +). Proof. It suffices to prove that the closure of C (R N +) W k,p (R N +) contains C (R N +) W k,p (R N +) (and use the Meyers-Serrin theorem). This is a consequence of the fact that, if u W k,p (R N +), then u ε u in W k,p, where u ε (x) = u(x, x N + ε) Exercise. Check the above fact. Hint: translations are continuous in L p, 1 p <. We will need later the following fact: assume that supp u [ 1, 1] N 1 [0, 1]. Then supp P u [ 1, 1] N. Step 2. The case where the support of u lies near Ω We may cover Ω with a finite collection Ω 0,..., Ω m of open sets s. t. : Ω 0 = Ω, and for each i 1 there is a C k diffeomorphism Φ i of Ω i onto [ 1, 1] N s. t. Φ i (Ω i Ω) = ( 1, 1) N 1 (0, 1), Φ i (Ω i (R N \Ω)) = ( 1, 1) N 1 ( 1, 0), 18 j=1

19 Φ i (Ω i Ω) = ( 1, 1) N 1 {0}. Let, for some i 1, u i W k,p (Ω i Ω) be compactly supported in Ω Ω i. Let v i = u i Φ 1 i and consider P v i. In view of Exercise 1.53, we have w i = (P v i ) Φ i W k,p (R N ), and w i is an extension of u i. Step 3. Construction of the global extension Consider a partition of unity (ζ i ) i=0,...,m subordinated to the covering Ω 0,..., Ω m. Let u i = ζ i u. Then (with the notations of Step 2) P u = m u 0 + w i has all the required properties. j=1 On the way, we proved the following 1.59 Corollary. Assume Ω bounded and of class C k. Let 1 p <. Then C k (Ω) is dense in W k,p (Ω) Corollary. With Ω, k, p as above, let U be an open neighbourhood of Ω. Then we may choose P s. t. supp P u U. Proof. Choose, in the proof of Theorem 1.54, Ω i U Remark. In what follows, smooth will be a loose notion adapted to the statements, or rather to their proofs. E. g., consider the following statement: Let Ω be smooth and bounded. Then W 1,1 (Ω) L N/(N 1) (Ω). The proof goes as follows: we prove that P u L N/(N 1) (R N ), and next we restrict P u to Ω. In order to work, the proof needs the extension theorem to apply. In this special case, smooth = C 1. We will not insist on the smoothness requirements; a dumb s rule is that C will always suffice. 1.4 Inequalities, embeddings Continuous embeddings 1.62 Theorem (Morrey). Assume that N < p <. Let Ω be smooth and bounded (or R N, or a half space). Then W 1,p (Ω) C α (Ω). Here, α = 1 N/p (0, 1) Definition. We will use the shorthand standard domain for Ω which is either smooth and bounded, or R N, or a half space. 19

20 Proof. It suffices to consider the case where Ω = R N and u Cc (R N ). Let B a ball of radius r containing the origin. Since, for x R N, we have u(x) u(0) x 1 1 u u(0) Cr1 N B 0 u(tx) dt, we find that 0 B 1 u(tx) dxdt = Cr 1 N 1 Cr 1 N+N/p t N/p N u L p (tb) dt Cr α u L p. 0 0 tb u(y) t N dydt Of course, the same holds if 0 is replaced by any other point. Let now x, y R N. Pick a a ball of radius x y containing both x and y. We find that u(x) u(y) C x y α u L p. On the other hand, by taking r = u L p, we find that u L p u(x) u + Crα u L p C u 1 N/p L p u N/p L. p B 1.64 Remark. This embedding is optimal in the sense that the exponent α cannot be improved. Optimality, here and in the next theorems, is obtained via a scaling argument. Here it is how it works: assume that W 1,p C α. Then this inclusion is continuous, by the closed graph theorem. Thus u(x) u(y) C x y α ( u L p + u L p), u W 1,p (R N ). Fix now some u and apply this estimate to u(λ ), λ > 0. It follows that u(x) u(y) C x y α (λ 1 N/p α u L p + λ N/p α u L p). If u is non constant and we take x, y s. t. u(x) u(y), we find, by letting λ, that α 1 N/p Remark. Let us take a closer look to the estimates we obtained in u(x) u(y), is x y α R N. The higher order term in the C α norm, namely sup x y estimated only by the higher order term in the W 1,p norm, namely u L p. The lower term, u L, is estimated by a portion of each term in the W 1,p norm. This is typical for all the embeddings of this kind, and can be guessed by the scaling argument Theorem. [Gagliardo-Nirenberg] Let Ω be a standard domain. Then W 1,1 (Ω) L N/(N 1) (with the convention 1/0 = ). 20

21 Proof. Assume Ω = R N and u Cc (R N ). For j = 1,..., N, we have xj u(x) = u (x 1,..., x j 1, t, x j+1,..., x N ) dt x j F j (x 1,..., x j 1, x j+1,..., x N ) := u (x 1,..., x j 1, t, x j+1,..., x N ) x j dt. R We find that ( 1/N u(x) G(x) := F j (x 1,..., x j 1, x j+1,..., x N )). (1.3) j Noting that F j L 1 u L 1, we conclude via the next result Lemma. Let F 1,..., F N L 1 (R N 1 ), and define G as in (1.3). Then G L N/(N 1) and G L N/(N 1) F j 1/N L. 1 j Proof. The cases N = 1, 2 are trivial. Assuming that the lemma holds for N, we proceed as follows: let x j := (x 1,..., x j 1, x j+1,..., x N+1 ). By the induction hypothesis, we have, for fixed x N+1 F 1 (x 1 ) 1/(N 1)... F N (x N ) 1/(N 1) dx 1... dx N N j=1 F j (x j ) 1/(N 1) L 1. On the other hand, Hölder s inequality implies that, for fixed x N+1, we have G(x) (N+1)/N dx 1... dx N F N+1 1/N L 1 ( N ) (N 1)/N F j (x j ) 1/(N 1) dx 1... dx N. Using again Hölder s inequality, we find that G N+1/N F N+1 1/N L F 1 1 (x 1 ) 1/N L... F 1 N (x N ) 1/N L dx 1 N+1 R N+1 R F N+1 1/N L F 1 1 1/N L... F 1 N 1/N L ; 1 j=1 this is equivalent to the statement of the lemma. 21

22 1.68 Theorem. [Sobolev] Assume that 1 < p < N. Let Ω be a standard domain. Then W 1,p (Ω) L Np/(N p) (Ω). Proof. We start by noting that the conclusion of Theorem 1.66 holds for u C 1 c. Let > 1 to be fixed later and let u C c (R N ). Then u C 1 c, so that u = u L N/(N 1) L N /(N 1) C u L 1 C u L p u 1. L ( 1)p (1.4) If we take s. t. N/(N 1) = ( 1)p, then we obtain that u L Np/(N p) C u L p, u Cc (R N ). We conclude as usual Definition. For 1 p < N, one usually denotes by p the exponent p = Np N p Theorem. Assume that N 2 and let Ω be a standard domain. Then W 1,N (Ω) L (Ω). However, we have W 1,N (Ω) L q (Ω), p q <. Proof. Assume, e. g., that Ω = B(0, 1). Let u(x) = ln x α. Here, 0 < α < 1 1/N. Then d u x 1 ln x α 1, by Proposition Our choice of α implies that u L and u W 1,N. Using (1.4) with p = N and = N (this choice of is made in order to have ( 1)p = N), we find that u L N 2 /(N 1) C 0 u 1/N u (N 1)/N. (1.5) L N L N We next use again (1.4), but this time we take s. t. ( 1)p = N 2 /(N 1), i. e., = N + 1. With the help of (1.5), we find that u L (N+1)N/(N 1) C 1 u α 1 u 1 α 1 for some α L N L N 1 (0, 1). We continue with = N + k, k = 2, 3,..., and find by induction that u L (N+k)N/(N 1) C k u α k u 1 α k for some α L N L N k (0, 1). Therefore, u L (N+k)N/(N 1) C k u W 1,N. Let now N q <. Then there is some large k s. t. N q < 1 (N + k)n/(n 1). If θ [0, 1) is s. t. q = θ N + 1 θ (N + k)n/(n 1), then, by Hölder s inequality, we have u L q u θ L N u 1 θ L (N+k)N/(N 1) C q u W 1,N. This basic embeddings give birth to many others, which are obtained by combining the above theorems. Rather then giving a long and uninformative list, let us rather give some examples. 22

23 1.71 Example. Let Ω be a standard domain in R 3. Then W 2,4 (Ω) C 1,1/4 (Ω). Indeed, if u W 2,4, then u W 1,4. It follows that u C 1/4 (Ω), so that u C 1,1/4 (Ω) Proposition. If kp = N and Ω is a standard domain, then W k,p (Ω) L q (Ω) for p q <. In the exceptional case k = N, p = 1, we also have W N,1 (Ω) C(Ω). However, the embedding W k,p (Ω) L (Ω) does not hold when k = 1, 2,..., N 1 and kp = N. Proof. Counterexamples to the embedding W k,p (Ω) L (Ω) are obtained by considering, as in the proof of Theorem 1.70, maps of the form x α ln x β (with appropriate α and β). The validity of the embeddings W k,p (Ω) L q (Ω) is established by induction on k (one has to apply Theorem 1.66 or 1.68 to D k 1 u). Finally, the embedding W N,1 C(Ω) is obtained, in the model case Ω = R N, from the identity u(x) = x1 valid for u C c (R N ). xn... N 1... N u(t 1,..., t N ) dt N... dt 1, Compact embeddings We start by noting that, once the embeddings in the preceding section established, they imply yet another family of embeddings, obtained via Hölder s inequality. For example: if N 2, then W 1,1 L N/(N 1) L 1, so that W 1,1 L p, 1 p < N/(N 1). We will call such an embedding suboptimal. Another example: in 2D, W 1,4 C 1/3. By contrast, the estimates established in the previous section will be referred as optimal Exercise. An optimal embedding cannot be compact. Hint: consider ϕ(n ) for a fixed ϕ Exercise. Assume that Ω = R N (or R N +). Then an embedding (optimal or not) cannot be compact. Hint: consider ϕ(ne 1 + ) for a fixed ϕ Theorem (Rellich-Kondratchov). Assume that Ω is smooth and bounded. Then suboptimal embeddings are compact. 23

24 Proof. We will not prove this theorem in all cases (the list is too long). However, we consider the two cases which lead to the general one. These cases are: Case 1. Assume that p > N. Let 0 < β < α = 1 N/p. Then W 1,p C β is compact Case 2. Assume that 1 p < N. Let 1 q < p = Np/(N p). Then W 1,p L q is compact Before starting, let us note that we may consider only functions supported in a fixed compact (this follows by the properties of the extension operator associated to a bounded domain). Case 1. In this case, we simply rely on W 1,p C α combined with 1.76 Lemma. Let U be bounded. Let 0 < β < α < 1. Then C α (Ω) C β (Ω) is compact. Proof. Let (u n ) be a bounded sequence in C α (Ω). By Arzelà-Ascoli, up to a subsequence we have u n u uniformly, for some u C α (Ω). We prove that u n u in C β (Ω). We may assume that u = 0. Then u n L 0. On the other hand { } 2 un L u n (x) u n (y) max x y x β y β, u n C α x y α β x y β. Thus u n C β { } un L o(1) + C max x y x y, u n β C α x y α β 0 as n. Case 2. This case relies on the following 1.77 Exercise. Let X be a complete metric space. Let (x n ) X. Assume that: for each ε > 0, there is a sequence (y n ) X s. t. d(x n, y n ) ε and (y n ) is relatively compact. Then the sequence (x n ) is relatively compact Exercise. We have u ρ ε W k,p u W k,p, ε > 0 and u C c (R N ). We consider a bounded sequence (u n ) W 1,p (U) s. t. supp u n K, where K U. By the two preceding exercises, we may further assume that u n Cc (R N ). Let, for fixed ε > 0, v n = u n ρ ε. Then (v n ) is bounded in C (R N ), and thus relatively compact in W 1,p (U), by Arzelà-Ascoli. It 24

25 remains to establish a uniform bound u n v n L q f(ε), where f(ε) 0 as ε 0. We drop the subscript n. We start with v(x) u(x) = (u(x εy) u(x))ρ(y) dy u(x εy) u(x) ρ(y) dy (1.6) C u(x εy) u(x) dy By integration, we find that C B(0,1) 1 B(0,1) 0 ε u(x tεy) dtdy. v u L 1 Cε u L 1. (1.7) Let now 1 q < p. Let θ (0, 1] be s. t. 1 q = θ p θ. Then p u v L q u v θ L1 u v 1 θ C u L p v θ L1 (u v) 1 θ L p Cε θ u θ L 1 u 1 θ L p Cεθ Equivalent norms As a starter, we recall the following 1.79 Exercise. Let u be a measurable function which is locally constant in the domain Ω. Then u is constant a. e. Hint: consider the set {x Ω; u = C a. e. in a neighbourhood of x} Proposition. Let Ω be a domain. Let u D (Ω) be s. t. u = 0. Then u is constant. Proof. It suffices to prove that u is locally constant. In particular, we may assume that Ω is a cube, say ( 2, 2) N, and prove that u is constant in ( 1, 1) N. Fix ψ Cc ( 2, 2) s. t. ψ = 1 in a neighbourhood of [ 1, 1]. We argue by induction (the case N = 1 is settled, cf Lemma 1.33). Define v through the formula v(η) = u(η(x 1,..., x N 1 )ψ(x N )) = u(η ψ), η C c (( 1, 1) N 1 ). 25

26 Clearly, v = 0, so that v = C. Let now ϕ Cc (( 1, 1) N ) and set η(x 1,..., x N 1 ) := ϕ(x) dx 1. Then ζ := ϕ η ψ Cc (( 2, 2) N ) and ζ dx 1 = 0. It follows that ζ = 1 for some Cc (( 2, 2) N. Consequently, u(ϕ) = u( 1 + η ψ) = v(η) = C(η) = C i. e., u = C Proposition (Poincaré). Let Ω be smooth bounded connected. Then u u + u L p is an equivalent norm on W 1,p. Proof. By the preceding proposition, the above quantity is a norm. Denote it by [ ]. It is easy to see that [u] C u W 1,p. Conversely, argue by contradiction: assume that there is a sequence (u n ) W 1,p s. t. u n + u n L p 1/n and u n W 1,p = 1. Then, up to a subsequence, u n u in L p, while u n 0 in L p. We find that u = 0, so that u is a constant. On the other hand, we have u n 0, so that u = 0. Therefore, u n 0 in W 1,p. This contradicts the fact that u n W 1,p = 1. If we apply the above proposition to u Poincaré s inequality ϕ, u, we find the usual form of 1.82 Corollary (Poincaré). Assume that Ω is smooth and bounded and let 1 p <. Then u u L p (Ω) C u L p (Ω). (1.8) 1.83 Lemma. Let Ω be a domain. Let u D (Ω) be s. t. D k u = 0. Then u is a polynomial of degree k 1. Proof. We know this for k = 0. We argue by induction. Since D k 1 j u = 0, there is some polynomial P j of degree k 2 s. t. j u = P j. Since k j u = j k u, we find that k P j = j P k. By the Poincaré lemma, in each ball B Ω there is some polynomial P (possibly depending on B) of degree k 1 s. t. j P = P j in B, j = 1,..., N. By the uniqueness principle for analytic maps, this P does not depend on B. Finally, we have (u P ) = 0, so that u = P + C. 26

27 1.84 Proposition. Let Ω be smooth and bounded. Let L : L p R M be a linear continuous functional s. t. Ker L does not contain any non zero polynomial of degree k 1. Then u Lu + D k u L p is an equivalent norm in W k,p. Proof. The proof is similar to the one of the Proposition The more difficult part: we argue by contradiction and obtain, via the compact embedding W k,p W k 1,p, the existence of a u W k,p s. t. u W k 1,p = 1, Lu = 0 and D k u = 0. Thus u = 0 (since u is a polynomial of degree k 1 and u is in the kernel of L). This contradicts the fact that u W k 1,p = 1. Similarly, we have the following result whose proof will be omitted Proposition. Let 1 p, q, r, 0 < m < k. Then u (m) L q (0,1) C( u L p (0,1) + u (k) L r (0,1)). The next result is different in nature (since we are not in a compact embedding situation) Proposition. The norm u u L p + D k u L p is an equivalent norm in W k,p (R N ) (or in R N +). Proof. This is a consequence of the following theorem (applied to p = q = r and 0 = l < m < k) Theorem (Gagliardo-Nirenberg). Let 1 p, q, r, 0 l m k be s. t. the following compatibility conditions m = θl + (1 θ)k, 1 q = θ p + 1 θ r (for some θ [0, 1]) are satisfied. Then, for u : R N R, we have D m u L q C D l u θ L p Dk u 1 θ L r C( Dl u L p + D k u L r). In particular, if u D (R N ) is s. t. D l u L p and D k u L r, then D m u L q Remark. Note that θ = k m, so that q is determined by the other k l parameters. More specifically, the compatibility condition is equivalent to requiring that q is given by 1 q = k m 1 k l p + m l 1 k l r. 27

28 Proof. The statement will be reduced to a special case. First reduction: assume that the theorem holds when u is smooth. Then it holds for every u. To see this, it suffices to note that, when u L s (even for s =!) we have u ρ ε L s u L s as ε 0. Second reduction: it suffices to know that the inequalities hold when l = 0. (Then apply the inequality not to u, but to D l u.) Third reduction: it suffices to prove the result when l = 0, k = 2. The general case is obtained by induction on k l. (See Exercise 1.91.) Thus we take l = 0, m = 1, k = 2. Fourth reduction, the most important one: it suffices to consider the case N = 1. Indeed, assuming the case N = 1 settled, we estimate j u, e. g., when j = N. Write R N x = (x, x N ). Then N u(x, ) L q C u(x, ) 1/2 L p 2 N u(x, ) 1/2 L. If we integrate this inequality w. r. t. r x and use Hölder s inequality with exponents 2p 2p and, then we find the desired q r estimate. The key fact is that the above exponents are conjugate to each other (check!). We have thus reduced the theorem to the following special case: if 1 p, q, r and 1 q = 1 2p + 1 2r, and if u C (R), then u L q C u 1/2 L p u 1/2 L r (this is the one dimensional case). Yet another reduction: it suffices to consider the case where R is replaced by R +. We may also assume that u L r = 1. Finally, we present the argument when p, r (and thus q) are finite. The adaptation to the remaining cases is straightforward. By the Proposition 1.85 (with k = 2, m = 1), we have v L q (0,1) C( v L p (0,1) + v L r (0,1)) for every v C ([0, 1]). By scaling (i. e., applying this to u(x + l )), we find that, for each interval I of length l, we have u L q (I) C(l α u L p (I) + l α u L r (I)) := C(A(l) + B(l)). (1.9) Here, α := 1 1 2r + 1 2p > 0. Fix some ε > 0. If we take a look at the quantities involved in (1.9), we see that, when l, we have A(l) 0 and B(l). Thus B(l) > A(l) for large l. We define a first l, say l 1, as follows: if A(ε) < B(ε), then we take l 1 = ε and I 1 = (0, ε); we say that this interval is of type I. Otherwise, pick the first l 1 > ε s. t. A(l 1 ) = B(l 1 ). In this case, we take I 1 = (0, l 1 ); this is an interval of type II. Then start again, but at l 1, not at the origin; call the new intervals I 2,.... Fix next another number, say L > 0. We stop the construction of intervals when these intervals cover (0, L). This is achieved after a finite number of steps (since each interval is of length ε). Let (0, L) I 1... I m. Note 28

29 the following: if I i is type I, then u L q (I i ) Cε α u L r (I i ). On the other hand, if I i is of type II, then u L q (I i ) C u 1/2 L p (I i ) u 1/2 L r (I i ). We find that u q L q (0,L) C ε αq u q L r (I i ) + C u q/2 L p (I i ) u q/2 L r (I i ). type I type II By Hölder s inequality with conjugate exponents 2p 2p and, we find that q r the second sum is at most C u q/2 L p u q/2 L. r In order to estimate the first sum, we consider two cases: a) if r > 1, then αq > 1 (check!) and we estimate the sum with C(L/ε+1)ε αq. (Here, we use the fact that we have at most L/ε + 1 intervals) b) if r = 1, then we estimate the sum with Cε αq u L 1 Cε αq. If, in these estimates, we let first ε 0, next L, then we obtain the desired inequality Exercise. There is a flaw in the above proof: it may happen that B(l) 0. Prove that, in this case, u = 0 in (0, ), and conclude Exercise. Prove the theorem in the remaining cases (where one or both of p, r are infinite) Exercise. The purpose of this exercise is to explain how to prove the Gagliardo-Nirenberg for arbitrary k, l, m. As explained in the proof of Theorem 1.87, we may assume that N = 1, l = 0, and k 3. a) Prove that the inequalities hold under the additional assumption that u Cc. Hint: let, for 0 j k, q j be defined by the formula 1 = 1 j/m + j/m q j p r. Start from u (j) L q j C u (j 1) 1/2 L q j 1 u (j+1) 1/2 L q j+1, j = 1,..., k 1, and proceed by induction on k b) Let now u be arbitrary. Set u ε = u ρ ε. If u L p and u (k) L r, prove that u (j) ε L p L, j N. Prove also that, in the special case where r = 1, we have in addition that ε (x) = 0 lim x u(k 1) c) Let ϕ C c (R) be s. t. 0 ϕ 1 and ϕ(0) = 1. By applying the Gagliardo-Nirenberg inequalities (compactly supported case) to x u ε (x)ϕ(δx), prove that these inequalities hold for u ε. Hint: consider separately the exceptional cases where r = 1 or q = 1 29

30 d) Conclude Definition. We let, for 1 p <, W 1,p 0 (Ω) := Cc (Ω) W 1,p. When p = 2, we write H0(Ω) 1 rather than W 1,p 0 (Ω) Theorem (Poincaré). Let Ω be bounded in one direction. Then u u L p is an equivalent norm on W 1,p 0 (Ω). Proof. The hypothesis is that there is some unit vector v and some finite number l > 0 s. t. for each w v, the set {t R; w + tv Ω} is contained in an interval of length l. Since the statement we want to prove is invariant by isometries (check!), we may assume that v = e N. Fix x R N 1 and u Cc (Ω). The support of u(x, ) is contained in some (a, b), with b a l. Since u(x, x N ) = xn a N u(x, t) dt, we find that u(x, ) L p C N u(x, ) L p, with C depending only on l and p. By integration, we find that u L p C u L p. By density, we find that the preceding estimate holds in W 1,p 0 (Ω) Exercise. Use the above argument combined with the Rellich-Kondratchov theorem in order to prove the following: Assume that Ω is smooth and bounded. Let k N, k 2. Then u u L p + D k u L p is an equivalent norm in W k,p W 1,p 0 (Ω). 1.5 Traces We discuss here the properties of the restrictions of Sobolev maps to hyper surfaces, e. g., to the boundary of a smooth domain. Note that giving a meaning to the value of a Sobolev map on a hypersurface requires some thought, since a priori such maps are only defined a. e. In what follows, we will state the results for general domains, but content ourselves to prove the results we state in the model case where Ω = R N +; we already explained how to obtain the case of a standard domain from the model case. We identify R N + with H = R N 1. We start with the following 1.95 Proposition. Let Ω be a standard domain and let Σ := Ω. Then the map u u Σ, initially defined from C (Ω) into C (Σ), extends uniquely by density to a linear map (called trace map) u tr u from W 1,p (Ω) into L p (Σ), for 1 p <. 30

31 1.96 Remark. It is easy to see that, when Ω is standard, we have tr W 1, (Ω) = Lip(Σ). Proof. It suffices to consider the case where Ω = R N + and u Cc (R N +). Fix a function ϕ Cc (R) s. t. ϕ(0) = 1 and supp ϕ ( 1, 1). If u Cc (R N +), then v = uϕ(x N ) Cc (R N +) and u H = v H. In addition, it is clear that v W 1,p C u W 1,p. It therefore suffices to prove that v H L p C v W 1,p. This follows from 1 p v(x, 0) p dx = N v(x, t)dt dx Dv p Dv p L p. H H 0 H (0,1) When 1 < p <, the above proposition is not sharp, in the following sense: if f is an arbitrary map in L p (R N 1 ), we cannot always find a map u W 1,p s. t. tr u = f. In other words, the trace map is not onto between the spaces we consider. Our next task is to determine the image of the trace map Definition. For 0 < s < 1 and 1 p <, we define RN W s,p = W s,p (R N ) = {f L p (R N f(x) f(y) p ) ; dx dy < }, x y N+sp equipped with the norm ( f W s,p = f L p + R N R N ) RN f(x) f(y) p 1/p dx dy. x y N+sp We let the reader check that W s,p is a Banach space. The main result of this section states that tr W 1,p (R N +) = W 1 1/p,p (R N 1 ) (and a similar result holds when Ω is standard). We start with some preliminary results Lemma. C (R N ) W s,p (R N ) is dense into W s,p (R N ) for 0 < s < 1 and 1 p <. Proof. Let ρ be a standard mollifier. We will prove that, if u W s,p, then u ε = u ρ ε u in W s,p as ε 0. Clearly, u ε u in L p. It remains to prove that, with v ε = u ε u, we have I ε = R N RN v ε (x) v ε (y) p x y N+sp dxdy = 31 R N RN v ε (x + h) v ε (x) p h N+sp dxdh 0.